# Tag Info

Accepted

### What does it mean for a set to have Lebesgue measure zero?

Let me quote von Neumann twice: There's no sense in being precise when you don't even know what you're talking about.   Young man, in mathematics you don't understand things. You just get ...
• 396k
Accepted

### Are most matrices diagonalizable?

Yes. Here is a proof over $\mathbb{C}$. Matrices with repeated eigenvalues are cut out as the zero locus of the discriminant of the characteristic polynomial, thus are algebraic sets. Some matrices ...
• 5,608
Accepted

### Apparent inconsistency of Lebesgue measure

"As the rationals are dense in $\mathbb{R}$ we must have $\mathbb{R}\subset \bigcup_{n=1}^\infty I_n$." This is false. Pick your favorite irrational number $x$. For every $n$, there exist infinitely ...
• 78.1k

### Intuitive, possibly graphical explanation of why rationals have zero Lebesgue measure

You could utilize one of the well known ways to count the rational numbers, namely consider the integer lattice $\mathbb Z^2$ and the subset $\{(a,b)\mid a\geq 1\ \wedge\ b\geq 0\}$ as illustrated ...
• 18.5k
Accepted

### Intuitive, possibly graphical explanation of why rationals have zero Lebesgue measure

This is a really hard question; I think in general intuition for this sort of thing tends to come with experience, as you get used to the concepts. Having said that, I'll try to articulate the way ...
• 3,921

### What does it mean for a set to have Lebesgue measure zero?

Practically speaking this means that when you integrate functions such sets don't matter, in the sense that if you modify the domain of integration by a set of measure zero, the integral is unchanged.
• 44k

### Intuitive, possibly graphical explanation of why rationals have zero Lebesgue measure

This isn't a geometric answer, but you can get a lot of intuition for Lebesgue measure by thinking about it probabilistically. Specifically: The measure of a subset $S\subseteq [0,1)$ is the same ...
• 49.6k

### Is the measure of the sum equal to the sum of the measures?

Another example: $$A=\mathbb{Z},\quad B=[0,1],\quad A+B=\mathbb{R}.$$
• 76.6k

### Prove that the graph of a measurable function is measurable.

Hint: Show that the mapping $$(x,y) \mapsto T(x,y) := f(x)-y$$ is measurable. Conclude that $\Gamma(f) = T^{-1}(\{0\})$ is measurable.
• 121k

### A possible paradox about topology and the relation between zero and infinity?

There are a couple of different issues going on here: if we increase the number of dimension ... the volume increases Here's the first problem: the volume of an $n$-dimensional cube whose side ...
• 5,252

### Measure Theory - Why doesn't empty interior imply zero measure?

You have no reason to assume that the measure of the boundary is $0$ too. And you provided an example yourself: the irrationals. Its boundary is $\Bbb R$, whose Lebesgue measure is infinity.
Accepted

### Is diameter of a set a measure?

Observe that the diameter of singletons is $0$ and the diameter of set $\{x,y\}$ is $d(x,y)>0$ if $x\neq y$. So there is no additivity.
• 151k